On higher Gauss maps

We prove that the general fibre of the $i$-th Gauss map has dimension $m$ if and only if at the general point the $(i+1)$-th fundamental form consists of cones with vertex a fixed $\mathbb P^{m-1}$, extending a known theorem for the usual Gauss map. We prove this via a recursive formula for expressing higher fundamental forms. We also show some consequences of these results.Comment: 12 pages, AMS-LaTeX; to appear in the Journal of Pure and Applied Algebr

Togliatti systems

We find some examples in P 5 (C) of surfaces satisfying Laplace equations. In particular, we study rational surfaces in P 5 (C) whose hyperplane sections have genus one that satisfy a Laplace equation. Then we study monomial Togliatti systems of cubics for variety of dimension three, i.e. we find all the monomial examples of three-folds satisfying Laplace equations

On the Hilbert vector of the Jacobian module of a plane curve

We identify several classes of curves $C:f=0$, for which the Hilbert vector of the Jacobian module $N(f)$ can be completely determined, namely the 3-syzygy curves, the maximal Tjurina curves and the nodal curves, having only rational irreducible components. A result due to Hartshorne, on the cohomology of some rank 2 vector bundles on $\mathbb{P}^2$, is used to get a sharp lower bound for the initial degree of the Jacobian module $N(f)$, under a semistability condition.Comment: 10 pages, 4 figures. To appear in Portugaliae Mathematic

Geometry of syzygies via Poncelet varieties

We consider the Grassmannian $\mathbb{G}r(k,n)$ of $(k+1)$-dimensional linear subspaces of V_n=H^0({\P^1},\O_{\P^1}(n)). We define $\frak{X}_{k,r,d}$ as the classifying space of the $k$-dimensional linear systems of degree $n$ on $\P^1$ whose basis realize a fixed number of polynomial relations of fixed degree, say a fixed number of syzygies of a certain degree. The first result of this paper is the computation of the dimension of $\frak{X}_{k,r,d}$. In the second part we make a link between $\frak{X}_{k,r,d}$ and the Poncelet varieties. In particular, we prove that the existence of linear syzygies implies the existence of singularities on the Poncelet varieties

Singular hypersurfaces characterizing the Lefschetz properties

In the paper untitled "Laplace equations and the Weak Lefschetz Property" the authors highlight the link between rational varieties satisfying a Laplace equation and artinian ideals that fail the Weak Lefschetz property. Continuing their work we extend this link to the more general situation of artinian ideals failing the Strong Lefschetz Property. We characterize the failure of SLP (that includes WLP) by the existence of special singular hypersurfaces (cones for WLP). This characterization allows us to solve three problems posed by Migliore and Nagel and to give new examples of ideals failing the SLP. Finally, line arrangements are related to artinian ideals and the unstability of the associated derivation bundle is linked with the failure of SLP. Moreover we reformulate the so-called Terao's conjecture for free line arrangements in terms of artinian ideals failing the SLP

Lefschetz Properties for Higher Order Nagata Idealizations

We study a generalization of Nagata idealization for level algebras. These algebras are standard graded Artinian algebras whose Macaulay dual generator is given explicity as a bigraded polynomial of bidegree $(1,d)$. We consider the algebra associated to polynomials of the same type of bidegree $(d_1,d_2)$. We prove that the geometry of the Nagata hypersurface of order $e$ is very similar to the geometry of the original hypersurface. We study the Lefschetz properties for Nagata idealizations of order $e$, proving that WLP holds if $d_1\geq d_2$. We give a complete description of the associated algebra in the monomial square free case.Comment: 16 pages, 4 figures. To appear in Advances in Applied Mathematic

Newton's lemma for differential equations

The Newton method for plane algebraic curves is based on the following remark: the first term of a series, root of a polynomial with coefficients in the ring of series in one variable, is a solution of an initial equation that can be determined by the Newton polygon. Given a monomial ordering in the ring of polynomials in several variables, we describe the systems of initial equations that satisfy the first terms of the solutions of a system of partial differential equations. As a consequence, we extend Mora and Robbiano’s Groebner fan to differential ideals

On the duals of smooth projective complex hypersurfaces

We show in this note that a generic hypersurface $V$ of degree $d\geq 3$ in the complex projective space $\mathbb{P}^n$ of dimension $n \geq 3$ has at least one hyperplane section $V \cap H$ containing exactly $n$ ordinary double points, alias $A_1$ singularities, in general position, and no other singularities. Equivalently, the dual hypersurface $V^{\vee}$ has at least one normal crossing singularity of multiplicity $n$. Using this result, we show that the dual of any smooth hypersurface with $n,d \geq 3$ has at least a very singular point $q$, in particular a point $q$ of multiplicity $\geq n$.Comment: v2. Theorem 1.4 is new and says something about the dual of any smooth hypersurface. Hence the slight change in the name of the paper. arXiv admin note: substantial text overlap with arXiv:2202.0223